4 Answers
4

There has been a lot of work on the spectrum of the operator $u+v + (u+v)^{\star}$. For example by Choi-Elliott-Yui (see here). Similar operators have also been studied from the Mathematical Physics community under the name Discrete Magnetic Laplacian, for example by Shubin (see here). There is also a lot more recent work. The spectrum turns out to be a Cantor set and the gaps can be studied in terms of the arithmetic of $\theta$.

However, I do not know much about $u+v$. Since it is not normal, this seems to be a slightly different story.

This is helpful information. We also should mention the paper of Nate Brown "AF embeddings and the numerical computation of spectra in irrational rotation algebras" which provides ways to check individual points for membership in the spectrum.
–
Jon BannonAug 17 '10 at 18:30

Perhaps it is helpful for you to know that you can find papers on $u + v + u^{\dagger} + v^{\dagger}$ also by looking for "Harper equation", "Discrete mathieu equation" or "Hofstadter butterfly".

Here's an example of the butterfly. Hoftstadter found the (rough) structure of the butterfly in 1976 by looking at a model for Bloch electrons (i.e. electrons in a periodic structure) in a magnetic field. The irrationality $\theta$ represents essentially the magnetic flux through a unit cell of the lattice.
(As for rational $p/q$ there is a translation symmetry one find $q$ "bloch bands", which touch at $E=0$ for pair $q$.)

I spent a part of my PhD thesis (no math, but renormalization from a more physical/heuristical point of view) on the multifractal properties of the spectrum for irrational values and gave some estimations on the minimal and maximal multifractal dimensions for quadratic irrationalities.
If you're interested, here's a paper of mine: http://iopscience.iop.org/0305-4470/30/1/009.

The following might be too naive, it comes from my understanding of the self-adjoint case. Consider the representation on $\ell^2(\mathbb{Z})$ given by
$$
U \psi(n) = e^{2\pi i n} \psi(n),\quad V\psi(n) = \psi(n+1).
$$
Then, we want to consider $H = U + V$. Now comes the thing, where I am not sure, if $z$ is in the spectrum, then there exists a solution $\varphi$ such that
$$
(H - z) \varphi = 0,\quad \lim_{n\to\pm\infty} \frac{1}{|n|}\log(|\varphi(n)|) = 0
$$
whenever $z$ is in the spectrum of $H$. We can assume that $\varphi(0) = 1$. A computation then shows that we must have
$$
\lim_{N \to \pm\infty} \frac{1}{|N|}\sum_{n=0}^{N} \log|z - e^{2\pi i n}| =0,
$$
which is the case whenever $|z| = 1$ (Here I use some ergodic theorem).